A very intriguing feature of the square- and cubic-root FSOTIs
studied here is that the non-Hermitian effect could induce more topological
corner modes than in the Hermitian limit. To demonstrate this, we
investigate the gap functions of \({\cal U}_{1/2}\)  and \({\cal U}_{1/3}\) 
versus the gain and loss strength \(v\)  in Figs. REF (a)
and REF (b). In both figures, \(v_{1}\)  and \(v_{2}\) 
denote critical values of \(v\)  where the system undergoes topological
phase transitions. Their specific values are determined by the gapless
condition of the parent model as shown in Eq. (REF ).
With the increase of \(v\)  from 0 to 2, we find that in the three
regions \(v\in (0,v_{1})\) , \((v_{1},v_{2})\) , \((v_{2},2)\) , the winding
numbers of the parent model are \((\nu _{0},\nu _{\pi })=(1,0)\) , \((3,0)\) ,
\((3,-2)\)  [1]}, whereas the number of corner modes are \((n_{\pi /2},n_{0},n_{\pi })=(4,0,0)\) ,
\((12,0,0)\) , \((12,8,8)\)  in Fig. REF (a) for the square-root
model \({\cal U}_{1/2}\) , and \((n_{0},n_{2\pi /3},n_{\pi /3},n_{\pi })=(4,4,0,0)\) ,
\((12,12,0,0)\) , \((12,12,8,8)\)  in Fig. REF (b) for the cubic-root system \({\cal U}_{1/3}\) . These results are clearly consistent with
the bulk-corner correspondence relations for square- and cubic-root
FSOTIs as stated in Eqs. (REF ) and (REF ).
